Questions tagged [sums-of-squares]

For questions concerning various representation of integers as sums of squares, which are studied in number theory.

108 questions with no upvoted or accepted answers
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11
votes
1answer
186 views

Standalone proof of a conditional part of Lagrange’s Four-Square Theorem?

Lagrange’s Four-Square Theorem — a special case of the Fermat-Cauchy Polygonal Number Theorem (FCPNT) — states that every natural number can be written as the sum of the squares of at most four ...
8
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0answers
229 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
7
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0answers
335 views

Yet an other conjecture about odd numbers $n=a+b$ such that $a^2+b^2$ is prime

This question is related to A conjecture about an unlimited path and Any odd number is of form $a+b$ where $a^2+b^2$ is prime but I present it on its own if anyone would like to help finding ...
7
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0answers
110 views

Applying iterated function on the sum of the squares of the prime factors of $30$

Let $f(n)$ denote the sum of the squares of the prime factors of $n$ with multiplicity. For example, $f(60)=f(2\cdot2\cdot3\cdot5)=2^2+2^2+3^2+5^2=42$. Denote the iterated function $f^k(n)=\...
6
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0answers
156 views

Which primes $\mathfrak{p}$ can be written as sums of two squares in $\mathbb{Q}(\sqrt[3]{2})$?

which primes $\mathfrak{p}$ can be written as sums of two squares in $\mathbb{Q}(\sqrt[3]{2})$? I am asking to solve an equation: $$ \mathfrak{p} = \big(a_1+a_2\sqrt[3]{2}+a_3\sqrt[3]{4}\big)^2 + \...
5
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0answers
115 views

Odd of the form $a^2+b^2+c^2+ab+ac+bc$.

The computation below shows that (for $a,b,c \in \mathbb{N}$) the form $$a^2+b^2+c^2+ab+ac+bc$$ covers every odd integer less than $10^5$ except those in $$I= \{ 5, 15, 23, 29, 41, 53, 59, 65, ...
5
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0answers
383 views

Probability of having at least $j$ collisions when tossing $m$ balls into $n$ bins

Suppose that we throw $m$ balls into $n$ bins uniformly and independantly at random. We consider collisions as distinct unordered pairs, e.g., if 3 balls are tossed in one bin, we count 3 collisions. ...
4
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0answers
83 views

Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...
4
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0answers
151 views

Iterations of $x^2 + y^2$

We construct a sequence $S$ of distinct positive integers as follows 1) the sequence $S$ starts as $1,2,3$ 2) If $x,y$ are in the sequence , then $x^2 + y^2 $ is also in the sequence. 3) the ...
4
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0answers
132 views

Even numbers has the form $a+b$ where $\frac{a^2+b^2}{2}$ is prime

Conjecture: All even numbers $n>2$ can be written $n=a+b$ where $a,b\in\mathbb N^+$ and $\frac{a^2+b^2}{2}\in\mathbb P$. Is tested for $n<10,000,000$. This conjecture is related to and maybe ...
4
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0answers
767 views

Gauss' “Eureka” theorem

Gauss proved that every integer is expressible as the sum of three triangular numbers. I was wondering if the proof is anywhere to be found?
4
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1answer
90 views

Prove $\forall n\in\mathbb{N}, \exists m\in\mathbb{N}; n=\pm1^2\pm2^2\pm\cdots\pm m^2.$

And we choose the positive and negative signs in a way that the equation becomes true. I think it can be proved with mathematical induction. So here's how I begin: For $n=1$, $1=+1^2$ which is true. ...
3
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0answers
67 views

Proof algorithm to find representations as sum of two squares

I saw in a book the following algorithm to find, given a prime $p\equiv 1 \pmod 4$, integers $a $ and $b $ such that $p=a^2+b^2$. Step 1 : Find an integer $0 <m <p$ such that $m^2\equiv -1\...
3
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1answer
93 views

Find the number of positive integer solutions to the equation $a^2+b^2 = p_1p_2p_3$

Find the number of positive integer solutions to the equation $$a^2+b^2 = p_1p_2p_3 $$ where the $p_i$ are distinct primes each congruent to $1$ mod $4$. My take: We can show each $p_i$ can be ...
3
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0answers
80 views

Express Integer as the sum of $k$ squares

I'm looking for a way to efficiently determine the solutions ( i.e. sets of $x_1^2, x_2^2, ..., x_k^2 $ ) that satisfy     $$n^2 = x_1^2 + x_2^2 + x_3^2 + ... + x_k^2 $$ $$x_1, x_2, ...,...
3
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0answers
1k views

How to prove Sum of squares of $n$ numbers is unique

I was solving a programming problem where I needed to prove that sum of squares of $n$ numbers is unique. i. e. if calculate the sum of squares of equal number of positive integers then if they are ...
2
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0answers
296 views

Can the sum of two squares be used to factor large numbers?

The method described below is different from Euler sum of two squares factorization method. The method below does not require 2 sum of two squares representations to factor a number. This post was ...
2
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0answers
226 views

Can the sum of two squares be used to determine if a number is square free?

Mathword (http://mathworld.wolfram.com/Squarefree.html) stated that "There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer."...
2
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1answer
34 views

Quasiconvexity of a specific quartic bivariate polynomial family

I have a quartic polynomial of shape $$(a\cdot x+b+c\cdot xy-d\cdot y+y^2\big)^2$$ where $a,b,c,d\in\mathbb Z_{>0}$ are coefficients. The polynomial is not convex. However is it quasiconvex for ...
2
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1answer
93 views

Find sum of squares of elements of QB

Given that Q is an orthogonal nxn matrix and B is an mxn matrix, how can we find the sum of squares of all elements of QB in terms of the sum of squares of all elements of B? I know that the sum of ...
2
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0answers
74 views

Sums of descending squares

I am interested in integers that can be expressed as a sum of squares. Specifically I am interested in integers that can be expressed as follows: $n=6*Sum (k^2+(k-a)^2+(k-2a)^2.....1^2)$ These ...
2
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0answers
85 views

Hard for-all-integer-problems proveable for very great integer limits

Computational experiments suggests the conjecture: All big enough odd numbers $N$ is of the form $N=\sum_{k=1}^n m_k$, where $\sum_{k=1}^n m_k^2$ is prime and all $m_k\in\mathbb Z^+$. Since the ...
2
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0answers
103 views

Sums of more than one combination of squares.

I'm interested in examples like these, where the sum of $n$ squares equals the sum of another $n$ squares. $3^2 + 11^2 \quad = \quad 7^2 + 9^2 \quad = \quad 130$ $5^2 + 6^2 + 10 ^ 2 \quad = \quad 4^...
2
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0answers
801 views

On the number of ways of writing an integer as a sum of 3 squares using triangular numbers.

It is well known that integers can be represented as a sum of squares, two, three and more. In what follows will be given a way to represent integers as a sum of 3 squares using triangular numbers. It ...
2
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0answers
59 views

Definite sum of a decreasing series

can we find a closed form for following sum: \begin{equation} \sum_{i=0}^{n} a^{i^2} = a^0 + a^1 + a^4+a^9 +...a^{n^2} \end{equation} where $0<a<1$ , what if $n \rightarrow \infty$ ?
2
votes
1answer
133 views

telescoping sum

I had to use the identity $\sin^{4}(x)=\sin^{2}(x)-\frac{1}{4}\sin^{2}(2x)$ to write the sum $\sum^\infty_{k=1} 4^k \sin^{4}(\frac{\theta}{2^k})$ as a telescoping sum. However, how would I use an ...
2
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0answers
35 views

Nonnegative vs SOS

Consider the polynomial $f(x_1, \cdots, x_n)$, I want to characterize $f$ being nonnegative, i.e., $f\geq 0$. For $n=1$, this is equivalent to saying that $f$ is SOS (sum of square). However, in ...
2
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0answers
652 views

Which integers are a sum of two relatively prime squares?

It's well known that a positive integer $n$ is a sum of two squares if and only if every prime of the form $4m + 3$ that divides $n$ appears with even multiplicity in the prime factorization of $n$. ...
2
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0answers
55 views

Polynomial Density Theorem

So I was consider Lagrange's 4-square theorem and came up with this generalization: Given a polynomial with rational coefficients $$P(x) = a_0 + a_1x + ... + a_nx^n$$ Determine if there exists ...
2
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0answers
133 views

Showing a Hermitian preserving map is not necessarily positivity preserving?

Say I have a linear map, which is not positivity preserving $$\phi: \mathscr H \to \mathscr H$$where $\mathscr H$ is the set of $n \times n$ Hermitian matrices. Then does there exist a positive ...
1
vote
1answer
28 views

Pattern in Squared Numbers and their Digit Sum

So this has been boggling my mind for some time now. On spring break I was really bored and started messing around with numbers when I noticed something. I was squaring each number (1-9) when I ...
1
vote
1answer
54 views

Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$

Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$. Given $m^2+n^2=0$ then $m^2= -n^2$. Because $m$ and $n$ are real numbers, then $m^2 \geq 0$, $n^2 \geq 0$. Therefore, $m=0$ and $n=0$. Is that ...
1
vote
1answer
33 views

Proof that every sum of exponents can be represented as a polynomial. I am missing an inital idea.

$$ s_n(p)=\sum_{k=1}^n k^p $$ Show: For every $q \geq 1$ exist rational numbers $ a_{k,q} , 1 \leq k \leq q-1 $, such that $$ s_n(q)= \frac 1 {q+1} n^{q+1}+ \frac 1 2 n^q + \sum_{k=1}^{q-1} a_{k,q}...
1
vote
1answer
106 views

Hypergeometric function 3F2 with unit argument

Recently I obtained the following expression $${}_3F_2(-n,a - b ,1-b-n; b + 1, 1-a-n; 1), $$ with $b>a>0$ and $n\in\mathbb{N}$. My question is: If someone knows a closed form solution to the ...
1
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0answers
93 views

Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way

Is there a formula that says if a number is the sum of 2 distinct nonzero squares ? This is the sequence I'm trying to emulate: 5, 10, 13, 17, etc.. http://oeis.org/A004431 (Invalid it contains 85, ...
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0answers
57 views

$\exists\infty$ many pairs of consecutive squares s.t. their sum is also a square

First of all, the term "pairs" is two of them, I assume (question's formulation is rather difficult to understand for me). So I guess this is the statement: $$\exists\infty\text{ many pairs of ...
1
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0answers
38 views

why sum of squares equals to assign the variable evenly in linear programming

I have a quesiton about linear programming. My objective is trying to make the utilization evenly. Example $U_{1}, U_{2},...,U_{k} $ $\sum_{i=1..k} U_{i} = C $ $C$ is some constant. $U_{i}$ is ...
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0answers
84 views

Can Gauss-Newton algorithm give better optimization performances than Newton algorithm?

I am currently facing the problem of a robotic manipulator calibration: the goal is to find the best correction that must be applied to a set of kinematic parameters describing the robot model, in ...
1
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0answers
159 views

How to interpret this visual proof for Archimedes' derivation of Sum of Squares?

I'm interested in knowing how was the first closed form solution of the sum of squares derived (for historical context/curiosity). I stumbled across an MAA Page that provides a visual proof, without ...
1
vote
1answer
38 views

Show that the number of solutions of $x^2+y^2 ≡ 1$ (mod $p$) where $0<x<p$, $0<y<p$, $p$ is odd prime is even iff $p ≡ 3, -3$ (mod $8$)

Show that the number of solutions of $x^2+y^2 ≡ 1$ (mod $p$) where $0<x<p$, $0<y<p$, $p$ is odd prime is even iff $p ≡ 3, -3$ (mod $8$). I learned about quadratic residue and sums of ...
1
vote
1answer
48 views

Looking for a reference about a problem on the number of representations of an integer as a sum of two squares

Few years ago I came accross a paper, or maybe a solution of a problem from a journal (possibly AMM or something like that) in which the following result was proved: The smallest positive integer ...
1
vote
1answer
29 views

Quadratic fit using sum of least squares without matrices

I've been hunting around for examples using the sum of least squares + partial derivative method to fit a polynomial to a set of points but am completely stuck. All the examples I've found involve ...
1
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0answers
88 views

The sum of the squares of $5$ consecutive primes is again prime. True for infinite many quintuples?

Let $(p,q,r,s,t)$ be a quintuple of consecutive primes. Are there infinite many such tuples such that $$p^2+q^2+r^2+s^2+t^2$$ is prime again ? Motivation : $p^2$ is never prime $p^2+q^2$ is ...
1
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0answers
56 views

Finding the pdf of sum of squared weighted gaussian variables

I have 3 sets of weighted gaussians that are part of 3 different Gaussian Mixture Models, phi1,phi2 and phi3. phi1 has n1 gaussian components with weights wj, mean mu_j and variance eta_j_squared, j ...
1
vote
1answer
62 views

How can i find number of ways presenting number as sum of no more than 4 squares

I am given a number and I have to find number of ways to present that number as sum of no more than 4 squares.For example $25$ can be presented as $1^2+2^2+2^2+4^2$, $3^2+4^2$ and $5^2$.
1
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0answers
96 views

Product of SOS polynomials in the Gram matrix form by odd degree monomials.

This might be elementary, but I'm struggling to write the product of a sums of squares polynomial and an odd power of a variable using its Gram matrix. Following Pablo Parrilo's notation, let $p(x,y):...
1
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0answers
35 views

Can someone show me a simple example of finding simple squared error in k-means?

I'm working on some homework and one of our assignments asks us to find the final mean of each cluster and the SSE (sum of squared error). I haven't been able to find anything online and was wondering ...
1
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0answers
166 views

Sum of two squares of an integer N, the simplest algorithm?

The algorithm is based on the following observation. An integer $b>a$ can be written as $b=a + k$ with a and k integers. So if we want to find a sum of 2 squares of an integer N, we start by ...
1
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0answers
133 views

Primes as sum of squares in finite field

Is there an algorithm to find integers such that the following holds true : ${a_1}^2 + {a_2}^2 \dots + {a_k}^2 \equiv p $ $(mod $ $n)$, where $p$ is a prime and $n$ is any general integer. By ...
1
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0answers
65 views

The largest “root-sum” in a four-square representation

Well-known is the fact that every natural number can be represented as the sum of four integer squares — this proposition is often called Lagrange’s four-square theorem. Perhaps less well-known is ...